For the feedback control system shown in the figure, the process transfer function is $G p(s)=\frac{1}{s(s+1)}$, and the complification factor of the power amplifier is $K \geq 0$. The design specifications required for the system, time constant is $1 \; \mathrm{sec}$ and a damping ratio of $0.707$.
- Find the desired locations of the closed-loop poles.
- Write down the required characteristic equation for the system. Hence determine the PD controller transfer function $G_{0}(s)$ when $K=1$.
- Sketch the root-locus for the system.